A geometric view of orbital element perturbations

Many systems commonly encountered in celestial mechanics and astrodynamics can be effectively modeled as a two-body problem subject to relatively small perturbations. Under these circumstances, the deviations from the two-body orbit due to the perturbative acceleration can be described in terms of time-variable orbital elements.
 
In this post I will follow the elegant geometric treatment of the perturbation of the orbital elements given in Chapter VIII of Moulton (1902), from which I borrowed the key figures. This approach is complementary to the analytical derivation, usually leading to Lagrange's planetary equations. Having its roots in the first-ever treatment of orbital perturbations, which appeared in Newton's Principia, the geometric picture emphasizes intuitive, qualitative understanding over quantitative results. 
 
We shall focus on elliptic (i.e., bound) orbits, and restrict the discussion to the five classical orbital elements that describe the shape and orientation of the orbit: the semi-major axis $a$, the eccentricity $e$, the inclination $i$ (relative to a fixed reference plane), the longitude of the ascending node $\Omega$ (measured in the reference plane from a fixed reference direction), and the argument of periapsis $\omega$ (measured in the orbital plane from the ascending node). We consider a resolution of the disturbing force (i.e., the non-Keplerian perturbation) into the three components $S$, $T$, and $N$. The first component is orthogonal to the orbital plane and positive when directed toward the reference plane; the other two (in-plane) components are tangential and normal to the trajectory, and are considered positive in the direction of the motion, and toward the interior of the ellipse, respectively.

The key insight to be gained from the geometric picture is that each component of the perturbation only affects certain specific orbital elements, and that these connections can be understood on the basis of intuitive arguments.

After discussing the effects of each component of the perturbation separately, I will illustrate the results by means of numerical simulations. 

 

Orthogonal component

It is evident that the orthogonal component cannot produce in-plane orbital changes; $a$ and $e$ are therefore not affected, while $\omega$ is only affected indirectly, because it is measured from the ascending node, and therefore varies in lockstep with $\Omega$, according to $\dot \omega|_S = - \cos i \, \dot \Omega$. 

 


Figure 1. Adapted from Moulton, F. R. (1902). An Introduction to Celestial Mechanics. Macmillan.

 

To understand the changes in $i$ and $\Omega$ we examine Figure 1, which shows two positive orthogonal perturbations $S_0$ and $S_1$ (i.e., directed toward the reference plane, $AB$), applied at $P_0$ and $P_1$, respectively. Under their action, the orbital plane changes from $P_0Q_0$, to $P_0P_1$, to $P_1Q_1$. Clearly, both perturbation cause $\Omega$ to decrease, while $i$ is increased by $S_0$ and decreased by $S_1$. In particular, if the perturbations are small and approximately symmetric with respect to the ascending node, $i_2 \approx i_0$. Similar considerations apply in the vicinity of the descending node. We conclude that under the action of an orthogonal perturbation that is symmetric with respect to the reference plane during the entire orbit, the inclination is approximately unchanged, while the line of the nodes regresses.


Tangential component

Intuitively, an in-plane perturbation does not change $\Omega$ nor $i$. The simplest way to understand the effect on the semi-major axis is to take a short break from a purely geometric view, and consider the basic relations:

$v^2 = \mu \left( \dfrac{2}{r} - \dfrac{1}{a} \right) \ \ \  \Rightarrow \ \ \ \dfrac{da}{a^2} \propto v \, dv$.

A positive tangential perturbation always increases the orbital speed $v$, and thus, according to the relation above, always increases $a$ as well. 

 

Figure 2. Adapted from Moulton, F. R. (1902). An Introduction to Celestial Mechanics. Macmillan.
 

We now turn to the variation of $\omega$, and consider Figure 2. A positive tangential perturbation does not change the direction of the motion; on the other hand, the distance from the primary focus $E$ and its position are also not changed, while we have just seen that the semi-major axis always increases. It follows that $r_2 = 2\, a - r_1$ is increased, and the secondary focus is moved from $E_1$ to $E_1'$. This change produces an increase of $\omega$ when $T$ is applied in the first half of the orbit (between $A$ and $B$), and conversely in the second half (between $B$ and $A$). 

The change in $E_1$ affects the eccentricity as follows. At periapsis, the secondary focus is displaced along the line of the apsides, and thus $EE_1$ and $2a$ are changed by the same amount, but since $EE_1 < 2a$, the eccentricity $e = \dfrac{EE_1}{2a}$ is increased. Similarly, at apoapsis, $2a$ and $EE_1$ are increased and decreased, respectively, by the same amount, and the eccentricity is decreased. At some locations between $A$ and $B$ the change in eccentricity is zero, and it is easy to show that these points are on the minor axis.

 

Normal component

Once again, since a normal perturbation acts in the orbital plane, it does not affect $i$ nor $\Omega$. It also does not affect $a$, since it can only change the direction of the motion, but not the orbital speed $v$. 


Figure 3. Adapted from Moulton, F. R. (1902). An Introduction to Celestial Mechanics. Macmillan.

 

The effect on the line of the apsides is shown in Figure 3. An instantaneous normal perturbation applied at point $P$ changes the local tangent to the orbit from $PT$ to $PT'$. Since, by a well-known property of the ellipse, the radii $r_1$ and $r_2$ form equal angles with the tangent, the radius vector to the empty focus changes from $r_2$ to $r_2'$. On the other hand, the length of $r_1$ and $r_2$ stay the same, because $a$ does not change. The empty focus thus shifts from $E_1$ to $E_1'$. The overall effect on the line of the apsides is to rotate it forward when $N$ is applied within the arc $LAK$, and backward in $KBL$.

Under the action of a normal perturbation the eccentricity changes from $e = \dfrac{EE_1}{2a}$ to $e' = \dfrac{EE_1'}{2a}$. It is easy to see from Figure 5 that the eccentricity decreases if the perturbation occurs from $A$ to $B$, and increases in the second half of the orbit.


Summary

The effect of the perturbation on the orbital elements are summarized below in tabular form. For the naming of the notable points on the orbit, consult Figure 4.


Figure 4. Adapted from Moulton, F. R. (1902). An Introduction to Celestial Mechanics. Macmillan.

 

Comparison with numerical simulations

To illustrate the results derived so far, I have run some numerical simulations using the N-body code REBOUND. In the simulations, an otherwise unperturbed Keplerian orbit is subjected to instantaneous perturbations in the orthogonal, tangential, and numerical directions, as detailed below. The same reference points used in the geometric derivations (the nodes, the periapsis, apoapsis, etc.) are also marked for clarity.

Figure 5. Effect of orthogonal perturbations.

 

Figure 5 shows the effect of orthogonal perturbations. The instantaneous perturbations are applied symmetrically with respect to the nodes (see bottom panel); this produces a regression of the nodes, mirrored by an advancement of the line of the apsides (since $\dot \omega \propto - \dot \Omega$); the inclination retakes approximately its initial value at the end of a full orbit, and $a$ and $e$ are unaffected. 

Figure 6. Effect of tangential perturbations.

 

Figure 6 shows the effect of tangential perturbations applied at several points along the orbit. As expected from the preceding discussion, $a$ always increases, and $i$ and $\Omega$ are unaffected; $\omega$ increases when the perturbation occurs in the arc $AB$, and decreases otherwise; $e$ increases when the perturbation occurs between $C$ and $D$, otherwise it decreases.

Figure 7. Effect of normal perturbations.


Finally, figure 7 shows the effect of normal perturbations. Note the decrease of $e$ when the perturbation occurs in the first half of the orbit, and vice versa, while $\omega$ decreases when the perturbation is applied from $K$ to $L$, and increases otherwise. As expected, $a$, $i$, and $\Omega$ are unaffected by $N$.

 

References

Moulton, F. R., An Introduction to Celestial Mechanics. London: Macmillan, 1902.


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